On quasi-reversibility solutions to the Cauchy problem for the Laplace equation: regularity and error estimates

¹ Laboratoire Poems, CNRS/ENSTA/INRIA, Ensta Paris, Université Paris-Saclay, 828, Boulevard des Maréchaux, 91762 Palaiseau, France
² INRIA/Centre de mathématiques appliquées, École Polytechnique, Université Paris-Saclay, Route de Saclay, 91128 Palaiseau, France

^* Corresponding author: laurent.bourgeois@ensta.fr

Received: 20 June 2019
Accepted: 15 October 2019

Abstract

We are interested in the classical ill-posed Cauchy problem for the Laplace equation. One method to approximate the solution associated with compatible data consists in considering a family of regularized well-posed problems depending on a small parameter ε > 0. In this context, in order to prove convergence of finite elements methods, it is necessary to get regularity results of the solutions to these regularized problems which hold uniformly in ε. In the present work, we obtain these results in smooth domains and in 2D polygonal geometries. In presence of corners, due to the particular structure of the regularized problems, classical techniques à la Grisvard do not work and instead, we apply the Kondratiev approach. We describe the procedure in detail to keep track of the dependence in ε in all the estimates. The main originality of this study lies in the fact that the limit problem is ill-posed in any framework.